On a beam equation in Banach spaces

Abstract

This paper is concerned with the existence and asymptotic behavior of solutions of the Cauchy problem for an abstract model for vertical vibrations of a viscous beam in Banach spaces. First is obtained a local solution of the problem by using the method of successive approximations, a characterization of the derivative of the nonlinear term of the equation defined in a Banach space and the Ascoli-Arzelà theorem. Then the global solution is found by the method of prolongation of solutions. The exponential decay of solutions is derived by considering a Lyapunov functional.

Keywords

• beam equation
• global solution
• asymptotic behavior